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Radio station WCCO in Minneapolis broadcasts at a frequency of . At a point some distance from the transmitter, the magnetic-field amplitude of the electromagnetic wave from WCCO is 4.82×10-11T. Calculate (a) the wavelength; (b) the wave number; (c) the angular frequency; (d) the electric-field amplitude.

Short Answer

Expert verified
  1. The wavelength of the electromagnetic wave is 361 m
  2. The wave number of the electromagnetic wave is 174×10-4rad/m.
  3. The angular velocity of the electromagnetic wave is 5.22×106rad/s.
  4. The amplitude of electric field is 144×10-4V/m.

Step by step solution

01

Step 1: Define frequency and write formulas.

The number of waves that moves a fixed point in unit time is known as frequency.

f=1Torω2π

Where, T = Time period in seconds and is angular frequency in .

The frequency, wavelength, and speed of light of any wave are related by the wavelength-frequency relationship.

The wavelength-frequency relationship is:

c=λf

Where, c is the speed of light which is equal to 3×108m/s, λis the wavelength and is the frequency.

The wavelength is inversely proportional to the frequency.

λ=cforvf

Where, v is velocity of wave in m/s.

The relation between wave number and wavelength is:

k=2πλ

Where, k is wavenumber in rad/m.

The formula used to determine the amplitude of electric and magnetic fields of the wave are:

Emax=2lε0CBmax=Emaxc

02

Determine the wavelength and wave number of wave.

Given that,f=830×103Hz

The wavelength of wave is:

λ=cf=3×108830×103=361m

The wave number of wave is:

role="math" k=2πλ=2π361=174×10-4rad/s

Hence, wavelength of the electromagnetic wave is 361 m and the wave number of the electromagnetic wave is 174×10-4rad/m.

03

Determine the angular frequency and amplitude of electric field.

The angular frequency of wave is:

ω=2πf=2π830×103=5.22×108rad/s

Given that,Bmax=4.82×10-11T

The formula used to determine the amplitude of electric field of the wave is:

Emax=cBmax=3×1084.82×10-11=144×10-4V/m

Hence, the angular velocity of the electromagnetic wave is 5.22×106rad/sand the amplitude of electric field is144×10-4V/m .

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